Plane-like minimizers and differentiability of the stable norm

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Plane-like minimizers and differentiability of the stable norm

Antonin Chambolle, Michael Goldman, Matteo Novaga

To cite this version:

Antonin Chambolle, Michael Goldman, Matteo Novaga. Plane-like minimizers and differentiability of the stable norm. Journal of Geometric Analysis, 2014. �hal-00694872v2�

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Plane-like minimizers and differentiability of the stable norm

A. Chambolle

^∗

, M. Goldman

^†

M. Novaga

^‡

Abstract

In this paper we investigate the strict convexity and the differentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always differentiable in totally irrational directions, while in other directions, it is differentiable if and only if the corresponding plane-like minimizers satisfying a strong Birkhoff property foliate the torus. We also discuss the issue of the uniqueness of the correctors for the corresponding homogenization problem.

1 Introduction

We wish to investigate here a conjecture raised by Caffarelli and De La Llave in [21] concerning the differentiability of the so-called stable norm (or minimal action functional) in geometric Weak KAM theory. When considering the area functional in the Euclidean space, it is a classical result that hyperplanes are minimizers under compact perturbations. Caffarelli and De La Llave [21] proved the existence of plane-like minimizers for more general integrands of

the form Z

∂^∗E

F(x, ν^E)dH^d−1(x) + Z

E

g(x)dx

whereF(x, ν) is periodic inx, convex and one-homogeneous inν and whereg is a periodic function. It is then possible to define the homogenized energy ϕ(p) which represents the average energy of a plane-like minimizer in the directionp. Our aim is to study the regularity properties of this functionϕ. Our main results are the following:

• Ifpis “totally irrational” then∇ϕ(p) exists.

∗CMAP, Ecole Polytechnique, CNRS, Palaiseau, France, email: antonin.chambolle@cmap.polytechnique.fr

†Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, email: goldman@mis.mpg.de

‡Dipartimento di Matematica, Universit`a di Padova, via Trieste 63, 35121 Padova, Italy, email: novaga@math.unipd.it

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• The same occurs for any p such that the plane-like minimizers satisfying the strong Birkhoff property (see Section 4.1) give rise to a foliation of the space.

• If there is a gap in this lamination and if (q1, . . . , qk) ∈ Z^d is a maximal family of independent vectors such thatqi·p= 0, then∂ϕ(p) is a convex set of dimensionk, and ϕis differentiable in the directions which are orthogonal to{q1, . . . , qk}. In particular ifpis not totally irrational thenϕis not differentiable atp.

• ϕ² is strictly convex.

We also discuss the uniqueness of the minimizers (see Theorem 4.23 and Appendix B).

Our approach is based on the cell formula forϕ, introduced by the first author and Thouroude in [23] (see (6)). This formula provides a characterization ofϕ as the support function of some convex setCwhich gives the expression of its subgradient. Sinceϕis a convex function, it is differentiable at a pointpif and only if ∂ϕ(p) is reduced to a point. The set C being made of the integral over the unit cell of the calibrations in the directionp, the problem of the differentiability ofϕreduces to the investigation of whether two different calibrations in a given directionpcan have or not a different mean. The idea is to prove that every calibration calibrates every plane-like minimizer satisfying the strong Birkhoff property (see Section 4.1) and is thus prescribed on the union of these sets which form a lamination of the space. If there is no gap in this lamination (i.e. if it is a foliation) then the value of any calibration is fixed everywhere and thus its mean is also prescribed andϕ is differentiable atp. Ifpis totally irrational, every gap of the lamination must have finite volume. This can be used to show that the integral over a gap of two different calibrations must coincide, implying again the differentiability ofϕ. For non totally irrational vectors, the gap will be of infinite volume and using different heteroclinic minimizers inside the gap, one can show that there exists two different calibrations with different means, proving thatϕis non-differentiable.

One of the interesting features of our work is the connections it makes between the plane-like minimizers constructed in [23] (as level sets of appropriate correctors in a periodic homogenization problem), and the class of recurrent plane-like minimizers satisfying the strong Birkhoff property (see Section 4.1). We also give a clear presentation of the notion of calibration, and clarify the structure of the subgradient of functionals with linear growth (see also [5]), which could be of independent interest. Among the by-products of our analysis we obtain a proof of Ma˜ne’s conjecture in this setting (see Theorem 4.23 and Appendix B).

In [9, 27] the authors study the differentiability properties of the stable norm of the homology classH_d−1(M,R) for compact orientable Riemannian manifoldsM. IfM is thed-dimensional torus with a Riemannian metric, the stable norm is exactly our function ϕ. In this case, our results generalize [9, 27] in the sense that we consider interfacial energies which do not necessarily come from a Riemannian metric, and we also allow the presence of a volume term.

In some sense, a contribution of our work is also a simplification of some of the arguments

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of [9, 27], which comes from the use of the cell formula (6).

The stable norm is a geometric analog of the minimal action functional of KAM theory whose differentiability has first been studied by Aubry and Mather [8, 31] for geodesics on the two dimensional torus. The results of Aubry and Mather have then been extended by Moser [34], in the framework of non-parametric integrands, and more recently by Senn and Bessi [37, 15].

In this context, the study of the set of non-selfintersecting minimizers, which correspond to our plane-like minimizers satisfying the Birkhoff property has been performed by Moser and Bangert [33, 13], whereas the proof of the strict convexity of the minimal action has been recently shown by Senn [38]. Another related problem is the homogenization of periodic Riemannian metrics (geodesics are objects of dimension one whereas in our problem the hypersurfaces are of codimension one). We refer to [20, 30, 19] for more information on this problem.

The plan of the paper is the following. In Section 2 we recall some well known facts about functions of bounded variation, sets of finite perimeter, convex anisotropies, pairings between BV functions and bounded vector fields, and introduce the notion of Class A minimizers, plane-like minimizers. We define the stable norm, which is, in fact, the homogenized surface tension of a periodic homogenization problem and can be recovered with a cell formula. In Section 3 we give some general properties of Class A Minimizers. In Section 4 we define the calibrations and prove an important regularity result (Theorem 4.6). In Section 4.1 we introduce the Birkhoff property and prove that every calibration calibrates every plane- like minimizer which satisfies this strong Birkhoff property. In Section 4.2 we construct heteroclinic surfaces in every gap and use them in Section 5 to study the strict convexity and differentiability properties ofϕ. In Section 6 we give some simple examples of energies for which the existence or absence of gaps can be proven. Finally in Section 7 we show that the set of stable norms that can be attained starting from isotropic energies is dense in the set of all symmetric anisotropies. Appendix A gives an elementary proof of a separation result in Z^d, while in Appendix B it is shown a generic uniqueness result for the correctors minimizing the cell formula.

Acknowledgments.

The second author acknowledges very interesting discussions with G. Thouroude and B.

Merlet. He also warmly thanks Carnegie Mellon University for its hospitality during the finalization of this work. The third author acknowledges partial support by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analysis, and control- theoretic problems”. Some of the main ideas of this work have been clarified during a fruitful stay of the first two authors at the University of Padova.

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2 Notation and main assumptions

2.1 Basic notation

We letQ^′ = (0,1)^d,Q= [0,1)^d and Tbe the torus (R^d/Z^d). For m∈N, we letH^m be the Hausdorff measure of dimensionm. Given a vectorp∈R^d we say that

• pis totally irrational if there is noq∈Z^d\ {0} such thatp·q= 0,

• pis not totally irrational if there exists such aq∈Z^d\ {0},

• pis rational ifp∈R·Z^d,

• pis irrational if it is not rational.

For a givenp∈R^dwhich is not totally irrational we let Γ(p) :={q∈Z^d : q·p= 0}.

Then, there exists (q1, . . . , qk) ∈ Γ(p) such that Γ(p) = SpanZ(q1, . . . , qk). By a Gram- Schmidt procedure we can find (¯q1, . . . ,q¯k) ∈ Γ(p) such that (¯q1, . . . ,q¯k) is an orthogonal basis ofV^r(p) := SpanR(q1, . . . , qk). Notice that in general SpanZ(¯q1, . . . ,q¯k)6= Γ(p). We let Vⁱ(p) := (Rp⊕V^r(p))^⊥ be the set of irrational directions, that is,Vⁱ(p)∩Z^d={0}. ForR >0 we let BR be the open ball of radiusR. Finally letS^d−1 be the unit sphere of R^d. In this paper we will take as a convention that all the sets are oriented by their inward normal.

2.2 BV functions

We briefly recall the definition of a function with bounded variation and a set of finite perimeter. For a complete presentation we refer to [4, 26].

Definition 2.1. Let Ω be an open set of R^d, we will say that a function u ∈ L¹(Ω) is a function of bounded variation if

Z

Ω|Du|:= sup

z∈C1 c(Ω)

|z|∞≤1

Z

Ω

udivz dx <+∞.

We denote byBV(Ω)the set of functions of bounded variation inΩ(whenΩ =R^d we simply writeBV instead ofBV(R^d)). We define similarly the set BV(T) of periodic functions of bounded variation by choosingΩ =Tin the above definition.

We say that a setE⊆R^d is of finite perimeter if its characteristic function χE has bounded variation. We denote its perimeter in an open set Ω by P(E,Ω) := R

Ω|DχE|, and write simply P(E)whenΩ =R^d.

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A function v ∈ BV(T) can equivalently be seen as a Q-periodic function, with R

T|Dv| = R

Q|Dv|(which is also equal toR

Q^′|Dv|iff|Dv|(∂Q) = 0, andR

Q+x|Dv|for a.e.x). We also letBVloc be the set of functions of locally bounded variation. Similarly we will say that a set is of locally finite perimeter if its characteristic function is inBVloc.

Definition 2.2. Let E be a set of finite perimeter and lett∈[0; 1]. We then define E^(t):=

x∈R^d : lim

r↓0

|E∩Br(x)|

|Br(x)| =t

. We denote by ∂E := E⁽⁰⁾∪E⁽¹⁾c

the measure theoretical boundary of E. We define the reduced boundary ofE by:

∂^∗E:=

x∈Spt(|DχE|) : ν^E(x) := lim

r↓0

DχE(Br(x))

|DχE|(Br(x)) exists and |ν^E(x)|= 1

⊆E(¹2). The vectorν^E(x)is the measure theoretical inward normal to the set E. When no confusion can be made, we simply denoteν^E by ν.

Proposition 2.3. If E is a set of finite perimeter then DχE = ν H^d−1 ∂^∗E, P(E) = H^d−1(∂^∗E)andH^d−1(∂E\∂^∗E) = 0.

An important link betweenBV functions and sets of finite perimeter is given by the Coarea Formula:

Proposition 2.4. Let u∈BV(Ω). For a.e.t∈R,{u > t} has finite perimeter and it holds

|Du|(Ω) = Z

RH^d−1(∂^∗{u > t} ∩Ω)dt.

2.3 Anisotropies

Let F(x, p) : R^d×R^d → R be a convex one-homogeneous function in the second variable such that there existsc0with

c0|p| ≤F(x, p)≤ 1

c0|p| ∀(x, p)∈R^d×R^d.

We say thatF is elliptic if for someδ > 0, the functionF −δ|p| is still a convex function.

We denote byW(x) :={p : F(x, p)≤1} the unit ball ofF(x,·) (whenF does not depend on the space variablexwe will just denote it byW). We define the polar function ofF by

F^◦(x, z) := sup

{F(x,p)≤1}

z·p

so that (F^◦)^◦ =F. If we denote byF^∗(x, z) the convex conjugate ofF with respect to the second variable then {F^∗(x, z) = 0}={F^◦ ≤1}. IfF(x,·) is differentiable then, for every p∈R^d,

F(x, p) =p· ∇^pF(x, p)

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and

z∈ {F^◦(x,·)≤1}withp·z=F(x, p) ⇐⇒ z=∇^pF(x, p).

IfF is elliptic andC²(R^d×R^d\ {0}) thenF^◦is also elliptic andC²(R^d×R^d\ {0}). Moreover for everyx, y, z∈R^d, there holds

F²(x, y)−F²(x, z)≥2 (F(x, z)∇^pF(x, z))·(y−z) +C|y−z|² (1) for some constantC not depending on x (and the same holds forF^◦). Inequality (1) just state that F² is uniformly convex. We refer to [36] for a proof of these results and much more about convex bodies.

2.4 Pairings between measures and bounded functions

We fix in the following an elliptic anisotropyF. Following Anzellotti [7] we define a generalized trace [z, Du] for functionsuwith bounded variation and bounded vector fields zwith divergence inL^d.

Definition 2.5. let Ω an open set with Lipschitz boundary, let u ∈ BV(Ω) and let z ∈ L^∞(Ω,R^d) withdivz∈L^d(Ω). We define the distribution [z, Du] by

h[z, Du], ψi=− Z

Ω

u ψdivz− Z

Ω

u z· ∇ψ ∀ψ∈ Cc^∞(Ω).

Ifu∈BV(T) and z∈L^∞(T,R^d), with divz∈L^d(T), we can easily define the distribution [z, Du] in a similar way.

Theorem 2.6. The distribution [z, Du] is a bounded Radon measure onΩ and if ν is the inward unit normal toΩ, there exists a function [z, ν] ∈L^∞(∂Ω)such that the generalized Green’s formula holds,

Z

Ω

[z, Du] =− Z

Ω

udivz− Z

∂Ω

[z, ν]u dH^d−1.

The function [z, ν] is the generalized (inward) normal trace ofz on ∂Ω. If u∈BV(T) and z∈L^∞(T,R^d), with divz∈L^d(T), there holds

Z

T

[z, Du] =− Z

T

udivz.

Given z ∈L^∞(T,R^d), with divz ∈ L^d(T), we can also define the generalized trace ofz on

∂E, whereE is a set of locally finite perimeter. Indeed, for every bounded open set Ω with smooth boundary, we can define as above the measure [z, DχE] on Ω. Since this measure is absolutely continuous with respect to|DχE|=H^d−1 ∂^∗Ewe have

[z, DχE] =ψz(x)H^d−1 ∂^∗E

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with ψz ∈ L^∞(∂^∗E;H^d−1) independent of Ω. We denote by [z, ν] := ψz the generalized (inward) normal trace of z on ∂E. If E is a bounded set of finite perimeter, by taking Ω stricly containingE, we have the generalized Gauss-Green Formula

Z

E

divz=− Z

∂^∗E

[z, ν]dH^d−1.

We notice that there has been a lot of interest in defining the trace of bounded vector fields with divergence a bounded measure, on boundaries of sets of finite perimeters [3, 24], since this is related to the study of conservations laws.

2.5 Plane-like minimizers and the stable norm

Given a setE of locally finite perimeter, we consider the energy E(E, A) :=

Z

∂^∗E∩A

F(x, ν^E)dH^d−1+ Z

E∩A

g(x)dx (2)

with F, g Q-periodic continuous functions, and R

Qg = 0. Here ν^E is the inner normal to

∂^∗E, so that R

∂^∗E∩AF(x, ν^E)dH^d−1 = R

AF(x, DχE). We also assume that F is convex, one-homogeneous with respect to its second variable, and satisfies for somec0>0

c0|p| ≤F(x, p)≤c⁻¹₀ |p| (3) for any (x, p)∈R^d×R^d. A fundamental assumption throughout the paper is that the energy is coercive, in the sense that

E(E, Q) ≥ δP(E, Q) (4)

for someδ >0 independent of E. This will be ensured if (3) holds andgis small enough in some appropriate norm. WhenF(x, p) =|p|and g= 0, the energyE is just the perimeter.

In that case, it is well known that planes are minimizers under compact perturbations. In addition, the Bernstein Theorem states that, ifd≤7, the only minimizers of the perimeter under compact perturbations are the hyperplanes (see [26]). In [21], Caffarelli and De la Llave proved that, for general energies E, even if hyperplanes are not minimizers anymore there still exist plane-like minimizers.

Definition 2.7. We say that a setE of locally finite perimeter is a Class A Minimizer ofE if, for anyR >0, the setE minimizes E(E, BR)under compact perturbation in BR. Theorem 2.8 ([21]). There exists M > 0 depending only on c0 and δ such that for every p∈R^d\ {0} anda∈R, there exists a Class A Minimizer E of E such that

x· p

|p| > a+M

⊆ E ⊆

x· p

|p| > a−M

. (5)

Moreover∂E is connected.

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Definition 2.9. If E satisfies (5)for some M >0, we say that E is a plane-like set. IfE is a Class A Minimizer ofE satisfying (5) we say thatE is a plane-like minimizer.

The existence of Class A Minimizers is closely related to the existence of minimizers of

functionals of the form Z

Rd

G(x, u,∇u)

satisfying sup|u(x)−p·x|<+∞for somep(the vectorpis often called the rotation vector) in Weak KAM theory (see [33, 37]). The analogous of the minimal action functional of Weak KAM theory in our setting is the so-called stable norm introduced by Federer.

Definition 2.10. Let p∈R^d\ {0} and let E be a plane-like minimizer ofE in the direction p. We set

ϕ(p) :=|p| lim

R→∞

1

ω_d−1R^d−1E(E, BR), whereωd−1 is the volume of the unit ball inR^d−1.

Caffarelli and De La Llave proved that this limit exists and does not depend onE. In [23], the first author and Thouroude related this definition to the cell formula:

ϕ(p) = min Z

T

F(x, p+Dv(x)) + Z

T

g(x)(v(x) +p·x)dx : v∈BV(T)

, (6) where the measureF(x, p+Dv) is defined forv∈BV(T) byF(x, p+Dv) :=F(x,_|p+Dv|^p+Dv )|p+

Dv|. It is obvious from (6) that ϕis a convex, one-homogeneous function. It is also shown in [23] that the minimizers of (6) give an easy way to construct plane-like minimizers:

Proposition 2.11([23]). Letvpbe a minimizer of (6)then for everys∈R, the set{vp(x)+

p·x > s}is a plane-like minimizer ofE in the direction p.

We will make the following additional hypotheses onF, g:

• F isC^2,α(R^d×(R^d\ {0})), andg∈ C^1,α(R^d),

• F is elliptic (that isF(x, p)−C|p|is a convex function ofp)

Under these assumptions, one can show that there exists a periodic vector field σ of class C^2,α, with divσ=gand such thatF^′(x, p) =F(x, p)−σ(x)·p≥c^′₀|p|, for somec^′₀>0. The proof follows the same idea as in [23] (the only difference is that thanks to our regularity assumptions we need not rely on [18]). From (4) and (3), it follows (see for example [23]) thatR

TF(x, Dv) +R

T(1 +ε)gv dx≥δ/2R

T|Dv| ifε >0 is small enough. Hence, (1 +ε)g∈

∂H(0), the subgradient at zero of H(v) := R

TF(x, Dv). This is a 1-homogeneous, convex, l.s.c. functional defined on L^p(T) for any p∈ [1,∞] (letting H(v) = +∞ for v 6∈ BV(T)), which is the support function of

KH ={divσ : σ∈ C^∞(T;R^d), F^◦(x, σ(x))≤1∀x∈T}.

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From Hahn-Banach’s theorem, one deduces that (1 +ε)g is in the closure of KH for the topology (L^p^′, L^p). For p < ∞, large, it coincides with the strong closure, hence (1 +ε)g is the L^p-limit of a sequence divσn, with σn smooth andF^◦(x, σn(x)) ≤1. We solve then

∆un=g−div (σn/(1 +ε)) in the torus. On one hand,un∈ C^3,α(T) by elliptic regularity. On the other hand,kunk^2,p≤Ckg−div (σn/(1 +ε))k^p, which goes to zero withn. In particular, k∇unk^∞is arbitrarily small. We choosenso large that this quantity is less thanc0ε/(2 + 2ε), and letσ=σn/(1 +ε) +∇un∈ C^2,α(T). Then for anyp,σ(x)·p≤F(x, p)/(1 +ε) +∇un·p≤ F(x, p)−c0ε/(1 +ε)|p|+c0ε|p|/(2 + 2ε), so that the claim holds withc^′₀=c0ε/(2 + 2ε)>0.

For this reason, we can replace without loss of generalityF with F^′ and g with zero in (2) without changing anything to the problem. To simplify the notation we will therefore assume thatg= 0 in the rest of the paper.¹

In the following we let

X :={z∈L^∞(T,R^d) : divz= 0, F^◦(x, z(x))≤1a.e}.

We remark thatX is closed (hence compact) for theL^∞weak-∗topology. Indeed ifzn ∈X, zn ∗

⇀ zone sees that for any p∈Q^d, the average of z·pin any ballBρ(x) is less than the average ofF(·, p) (since it is true forzn). For a.e.xit follows thatz(x)·p≤F(x, p) for all p∈Q^d (henceR^d), that is, F^◦(x, z(x))≤1.

The following characterization of the subdifferential of one-homogeneous functionals is classical and readily follows for example from the representation formula [17, (4.19)] (see also [23, Prop 3.1].

Proposition 2.12. A functionv ∈BV(T) is a minimizer of (6) if and only if there exists z∈X such that

[z, Dv+p] =F(x, Dv+p).

The next result is the starting point of our analysis on the differentiability properties ofϕ.

Proposition 2.13. There holds

ϕ(p) = sup

z∈X

Z

T

z

·p .

Proof. This is a standard convex duality result. Notice that the sup on the right-hand side

1The hypothesisg ∈ C^1,α could be relaxed to g Lipschitz. Indeed, the regularity hypothesis onF is mainly there to ensure that regularity theory and maximum principle hold for the plane-like minimizers (see Proposition 3.4).

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is in fact a max, sinceX is compact. Now, for everyv∈BV(T) and everyz∈X, Z

T

[Dv+p, z] ≤ Z

T

F(x, Dv+p) hence

− Z

T

vdivz+ Z

T

z

·p ≤ Z

T

F(x, Dv+p) and since divz= 0, Z

T

z

·p ≤ Z

T

F(x, Dv+p) Z

T

z

·p ≤ϕ(p) taking the infimum onv, thus maxz∈X

Z

T

z

·p ≤ϕ(p).

To prove the opposite inequality letvp∈BV(T) be a minimizer in the definition ofϕ(p) and letz∈X be such that Z

T

[Dvp+p, z] = Z

T

F(x, Dvp+p), then

ϕ(p) = Z

T

F(x, Dvp+p)≤max

z

Z

T

z

·p.

Proposition 2.13 shows thatϕis the support function of the convex set C :=

Z

Q

z(x)dx : z∈X

, (7)

so that

ϕ(p) = max

ξ∈C ξ·p.

Observe thatCis trivially compact inR^d, beingX a compact set.

The subgradient ofϕat p∈R^d is given by

∂ϕ(p) = {ξ∈C : ξ·p=ϕ(p)}. (8) Anyξ∈∂ϕ(p) is associated to a fieldz as in (7). We will exploit the following fact:

ϕis differentiable atp ⇐⇒ ∂ϕ(p) is a singleton.

3 Properties of Class A Minimizers

We first start by recalling some well known facts about Class A and plane-like minimizers [2, 21, 23].

Proposition 3.1. LetEbe a Class A Minimizer. Then the reduced boundary∂^∗E is of class C^2,α andH^d−3(∂E\∂^∗E) = 0. Moreover, there existsγ >0andβ >0such that

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• if x∈E then|Br(x)∩E| ≥γr^d for everyr >0,

• if x∈E^c then|Br(x)\E| ≥γr^d for every r >0,

• if x∈∂E then βr^d−1≤ |DχE|(Br(x))≤β¹r^d−1 for every r >0.

As a consequence, we will assume in the following that our Class A Minimizers are all open sets (indeed, we can identify them with their points of density 1, which clearly are an open set because of the second density estimate). Moreover, the topological boundary ∂E agrees with the measure theoretical boundary ofE.

The stability of plane-like minimizers under convergence is a crucial point in the theory.

Proposition 3.2. LetEnbe a sequence of plane-like minimizers satisfying (5)with a uniform M and converging in the L¹_loc topology to a set E, then E is also a plane-like minimizer.

Moreover,En→E,¯ E_n^c →E^c, and of∂En→∂E in the Kuratowski sense.

Proof. The stability of the plane-like minimizers is a well known fact [21, Section 9]. The Kuratowski (or local Hausdorff) convergence easily follows from the uniform density estimates for plane-like minimizers (Proposition 3.1). Indeed, let ε > 0 be fixed and let x ∈ E∩ {y : d(y, ∂E)> ε}. Ifxis not inEn then by the density estimates we have

|En∆E| ≥ |Bε(x)\En| ≥γε^d.

This is impossible if n is big enough because |En∆E| tends to zero. Similarly, we can show that forn big enough, all the points ofE^c∩ {y : d(y, ∂E)> ε} are outside En. This shows that∂En ⊆ {y : d(y, ∂E)≤ε}. InvertingEn and E, the same argument proves that

∂E⊆ {y : d(y, ∂En)≤ε}giving the Kuratowski convergence of∂En to∂E.

Another simple (and classical) consequence of the density estimates is the following Proposition 3.3. Let u∈BVloc(R^d) then for everyR >0,

Spt(|Du|)∩BR = BR∩[

s

∂^∗{u > s}.

where in the union we consider only the levels for which{u > s} has finite perimeter inBR. If in additionvp is a minimizer of (6)andu(x) =vp(x) +p·x, then

• P({u > s} ∩BR)<+∞for every s∈R;

• ∂^∗{u > s}=∂{u > s};

• the function u⁺ is u.s.c., and u⁻ is l.s.c.;

• Spt(|Du|)∩BR= S

s∈R∂{u > s} ∩BR

∪ S

s∈R∂{u≥s} ∩BR .

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Here,u^±(x) are classicaly defined as the approximate upper and lower limits ofuatx, see [4].

Proof. Let us show the first assertion. Foru∈BVloc(R^d), ifx /∈Spt(|Du|), then there exists ρ >0 withBρ(x)⊆ {Du= 0}and thusuis constant onBρ(x), which implies that

x /∈[

s

∂^∗{u > s} ∩BR.

On the contrary, ifx∈Spt(|Du|) then for everyρ >0, by the Coarea Formula,

|Du|(Bρ(x)) = Z

RH^d−1(∂^∗{u > s} ∩Bρ(x))ds >0

thus for everyρ >0 there existsxρ ∈Bρ(x)∩∂^∗{u > sρ} for somesρ sincexρ tends to x whenρ→0, this proves the other inclusion.

Given a minimizervp, the other properties follow easily from the density estimates.

The following maximum principle for minimizers is a cornerstone of the theory.

Proposition 3.4. Let E1 ⊆E2 be two Class A Minimizers with connected boundary, then H^d−3(∂E1∩∂E2) = 0.

Proof. We shall now prove that ∂^∗E1 ∩∂^∗E2 = ∅. Let us assume by contradiction that

∂^∗E1∩∂^∗E26=∅then we can find ¯x∈∂^∗E1∩∂^∗E2such that∂^∗E1∩Br(¯x)6=∂^∗E2∩Br(¯x) for everyr >0. Since E1⊆E2,E1 andE2 have the same tangent space at ¯xand they can be seen as graphs over the same domainD of two functionsv1, v2∈C^2,α(D), with v2≥v1. For (y, r, p)∈R^d−1×R×R^d−1, let

F(y, r, p) :=e F((y, r),(−p,1)).

The functionsvi∈C^2,α(D),i= 1,2, (locally) minimize the functional Z

D

Fe(y, u,∇u)dy

and thus solve the elliptic PDE with H¨older continuous coefficients

∂Fe

∂r(y, vi,∇vi)−div [∇^pF(y, ve i,∇vi)] = 0. (9) Consider the functionw=v2−v1. Up to reducing the domain, we can assume that ¯x∈∂D, w >0 inDandw(¯x) = 0. We must then have∇w(¯x) = 0. Let

A(x) :=

Z 1

0 ∇²pFe(x, v2(x),∇v2(x)− ∇v1(x))dt B1(x) :=

Z 1 0 ∇^p ∂

∂rF(x, tve 2(x) + (1−t)v1(x)),∇v2(x))dt B2(x) :=

Z 1 0 ∇^p ∂

∂rF(x, ve 2(x), t∇v2(x) + (1−t)∇v1(x))dt c1(x) :=

Z 1 0

∂²

∂r²Fe(x, tv2+ (1−t)v1,∇v2)dt

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thenwsatisfies the linear non-degenerate elliptic PDE,

−div (A(x)∇²w)−div (B1(x)w) +B2(x)· ∇w+c1(x)w= 0.

By Hopf’s Lemma [25, Lemma 3.4], this implies that∇w(¯x)·νD<0, which gives a contradiction. Thus∂E1and∂E2can only intersects in singular points which are of (d−3)-Hausdorff measure zero [2].

Remark 3.5. In the case of isotropic functionals i.e. F(x, ν) =a(x)|ν|, [39] shows that in fact two minimizers which are contained one in the other cannot touch at all.

Proposition 3.6. Let d = 2 and let E be a Class A Minimizer, then E is a plane-like minimizer and∂E is connected.

Proof. By Proposition 3.4,∂E is of classC^2,α and is composed of a locally finite union of curves. Moreover each of these curves has infinite lenght by the minimality ofE. Letγ be such a curve andH be a plane-like minimizer with connected boundary. Due to the regularity and minimality ofE and H, ∂H can intersectγ in at most one point. Since by Theorem 2.8 there exist such plane-like minimizers inside every strip of width M, it follows thatγ is included in such a strip. Using again the minimality ofE, we then get that∂E is connected andE is plane-like.

Proposition 3.6 is reminiscent of Bernstein Theorem, and the same result also holds ford= 3 under the additional assumption that F does not depend on x [41]. However, in [32] it is shown that it is non longer true in four dimensions, even for a functionF independent ofx.

4 Calibrations

We now introduce the notion of calibration.

Definition 4.1. We say that a vector field z ∈ X is a periodic calibration of a set E of locally finite perimeter if, for every open setA, we have

Z

A

[z, DχE] = Z

A

F(x, DχE).

When no confusion can be made, by calibration we mean a periodic calibration.

The constant interest towards calibrations in the study of minimal surfaces, comes from the following result:

Theorem 4.2. If E is a set for which there exists a calibration (not necessarily periodic), thenE is a Class A Minimizer.

Proof. It follows by integration by parts, using divz= 0.

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Calibrations are very stable objects, as shown by the following Proposition:

Proposition 4.3. Let En be Class A minimizers converging in theL¹_loc-topology to a setE.

Assume that the setsEn are calibrated by zn∈X and thatzn converges to a fieldz weakly-∗ inL^∞. Thenz calibrates E (which is thus also a minimizer).

Proof. Letϕ∈ Cc^∞(R^d),ϕ≥0. Observe that sinceX is compact,z∈X. Hence Z

∂^∗En

ϕF(x, ν^Eⁿ)dH^d−1 = Z

Rd

ϕ[zn, DχEn] = − Z

En

zn· ∇ϕ

n→∞−→ − Z

E

z· ∇ϕ = Z

R^d

ϕ[z, DχE] ≤ Z

∂^∗E

ϕF(x, ν^E)dH^d−1 and the reverse inequality follows by lower-semicontinuity of the total variation. Hencez is a calibration forE.

A natural way of producing calibrations for a set is through the cell problem (6). By Propo- sition 2.13 there existsz∈X such that, for any minimizervp of (6),

Z

T

[z, Dvp+p] = Z

T

F(x, p+Dvp) = ϕ(p). (10)

We say that such a vector fieldzis a calibration in the directionp. Conversely, ifz∈X and v∈BV(T) are such that

Z

T

[z, Dv+p] = Z

T

F(x, p+Dv), (11)

then v is a minimizer of (6) and z is a calibration in the direction p. Repeating almost verbatim the proof of the Coarea Formula [4, Th. 3.40], there holds,

Proposition 4.4. Let A ⊆R^d be an open set. For u∈ BV(A), letting for s ∈ R, Es :=

{u > s}, there holds Z

R

Z

A

F(x, DχEs)ds= Z

A

F(x, Du).

We then deduce,

Proposition 4.5. Letz∈X be a calibration in the directionp, then for every minimizervp

of (6)and everys∈R,z calibrates the setEs:={vp+p·x > s}. Conversely forv∈BV(T), ifz∈X calibrates all the setsEs thenv is a minimizer of (6).

Proof. If z is a calibration in the direction p then as noticed above, z calibrates all the solutionvpof (6). Letvpbe one of these solutions then by the Coarea Formula and [7, Prop.

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2.7], for every Borel setA⊆R^d Z

R

Z

A

F(x, DχEs)ds= Z

A

F(x, Dvp+p)

= Z

A

[z, Dvp+p]

= Z

R

Z

A

[z, DχEs]ds

≤ Z

R

Z

A

F(x, DχEs)ds

and thus for almost every s ∈ R, z calibrates Es. Since for everys, Es = ∪^s^′^>sEs^′, by Proposition 4.3,z calibrates in fact everyEs. The converse implication follows by the same argument.

Theorem 4.6. Let E be a Class A Minimizer, letz∈X and letx¯ be a Lebesgue point ofz.

Then, ifz calibratesE and if x¯∈∂E, we have thatx¯∈∂^∗E and

z(¯x) =∇^pF(¯x, ν^E(¯x)). (12) Proof. Lettingzρ(y) =z(¯x+ρy), the assumption that ¯xis a Lebesgue point ofzyields that zρ→z¯inL¹(BR), hence also weakly-∗inL^∞(BR), for anyR >0, where ¯z∈R^dis a constant vector.

As usual, we letEρ:= (E−x)/ρ¯ and we observe thatEρ minimizes Z

∂Eρ∩BR

F(¯x+ρy, ν^E^ρ(y))dH^d−1(y)

with respect to compactly supported perturbations of the set (in the fixed ball BR). In particular, the setsEρ (and the boundaries∂Eρ) satisfy uniform density bounds, and hence are compact with respect to both localL¹and Hausdorff convergence.

Hence, up to extracting a subsequence, we can assume thatEρ→E, with 0¯ ∈∂E. Proposi-¯ tion 4.3 shows that ¯zis a calibration for the energyR

∂E∩B¯ RF(¯x, ν^E^¯(y))dH^d−1(y), and that E¯ is a plane-like minimizer calibrated by ¯z.

It follows that [¯z, νÊ^¯] = F(¯x, νÊ^¯(y)) for H^d−1-a.e. y in ∂E, but since ¯¯ z is a constant, we deduce that ¯E={y·ν¯≥0} with ¯ν=F(¯ν)∇^pF^◦(¯x,z). In particular the limit ¯¯ E is unique, hence we obtain the global convergence ofEρ→E, without passing to a subsequence.¯ We want to deduce that ¯x∈∂^∗E, with νÊ(¯x) =F(¯x, νÊ(¯x))∇^pF^◦(¯x,z), which is equivalent¯ to (12). The last identity is obvious from the arguments above, so that we only need to show that

ρ→0lim

DχEρ(B1)

|DχEρ|(B1) = ¯ν . (13) Assume we can show that

ρ→0lim|DχEρ|(BR) = |DχE¯|(BR) = ωd−1R^d−1

(14)

(17)

for anyR >0, then for anyψ∈C_c^∞(BR;R^d) we would get 1

|DχEρ|(BR) Z

BR

ψ·DχEρ = − 1

|DχEρ|(BR) Z

BR∩Eρ

divψ(x)dx

−→ − 1

|DχE¯|(BR) Z

BR∩E¯

divψ(x)dx = 1

|DχE¯|(BR) Z

BR

ψ·DχE¯

and deduce that the measureDχEρ/(|DχEρ|(BR)) weakly-∗converges toDχE¯/(|DχE¯|(BR)).

Using again (14)), we then obtain that

ρ→0lim

DχEρ(BR)

It remains to show (14). First, we observe that, by minimality ofEρand ¯Eplus the Hausdorff convergence of∂Eρ in balls, we can easily show the convergence of the energies

ρ→0lim Z

∂Eρ∩BR

F(¯x+ρy, ν^E^ρ(y))dH^d−1(y) = Z

∂E∩B¯ R

F(¯x, ν^E^¯(y))dH^d−1(y) (16) and, by the continuity ofF,

ρ→0lim Z

∂Eρ∩BR

F(¯x, ν^E^ρ(y))dH^d−1(y) = Z

∂E∩B¯ R

F(¯x, ν^E^¯(y))dH^d−1(y). (17) Then, (13) follows from a variant of Reshetnyak’s continuity theorem where instead of using the Euclidean norm as reference norm, we use the uniformly convex normF(¯x,·). Notice that this variant is covered by the original version of Reshetnyak [35]. For the reader’s convenience we sketch the proof here, following very closely the proof of [4, Thm. 2.39]. In what follows, the point ¯xis fixed and thus we will not specify the dependence of the functions on ¯x (for exampleF(p) will stand forF(¯x, p)).

Let nowµρ :=νÊ^ρH^d−1 ∂^∗Eρ, µ:=νÊ^¯H^d−1 ∂^∗E,¯ θρ := _F(ν^νÊρ_Eρ₎, θ = _F(ν^νÊ^¯E¯) and W :=

{F(p)≤1}. Then we define the measures ηρ onBR×∂W by settingηρ:=F(µρ)⊗δθρ(x). The sequenceηρ is bounded and thus there exists a weakly-∗ converging subsequence to a measureη. Letπ:BR×∂W →BR be the projection, thenF(µρ) =π#ηρ and thus by [4, Rk. 1.71], F(µρ) weakly-∗ converges to π#η, therefore by (17) and [4, Prop. 1.80] we get π#η=F(µ).

By the Disintegration Theorem [4, Th. 2.28], there exists aF(µ)-measurable map x→ηx

such thatηx(∂W) = 1 and η=F(µ)⊗ηx. Arguing exactly as in [4, Th. 2.38], we have Z

∂W

ydηx=θ(x) forF(µ)−a.e. x∈BR. (18) The anisotropic ballW being strictly convex andθ(x) being on its boundary, this will imply that indeed, ηx =δθ(x) which will conclude the proof. Since F is strictly convex, for every

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y∈∂W, (1), yields F(y)²−F² ν^E^¯

F(ν^E^¯)

!

≥2F ν^E^¯ F(ν^E^¯)

!

∇F ν^E^¯ F(ν^E^¯)

!!

· y− ν^E^¯ F(ν^E^¯)

! +C

y− ν^E^¯ F(ν^E^¯)

2

,

from which it follows

2 1− ∇F ν^E^¯ F(ν^E^¯)

!!

·y

!

≥C

y− ν^E^¯ F(ν^E^¯)

2

.

Integrating this inequality on∂W and using (18) we get

0 = 2 1− ∇F ν^E^¯ F(ν^E^¯)

!!

· Z

∂W

y dηx

!

≥C Z

∂W

y− ν^E^¯ F(ν^E^¯)

2

henceηx=δθ(x). The proof of (14) now easily follows. Indeed, sinceηρ(BR×∂W) converges toF(µ)(BR) =η(BR×∂W), using [4, Prop 1.80] we find

ρ→0lim Z

|θρ|(x)dF(µρ)(x) = Z

BR×∂W|θρ(x)|dηρ(x, y)

= Z

BR×∂W|y|dη(x, y) = Z

BR

|θ(x)|dF(µ)(x).

Remark 4.7. In the isotropic casea(x)|ν|, Auer and Bangert proved a similar result [9, Th.

4.2]. In that case, the monotonicity formula directly implies the convergence of the blow-up to a cone which is calibrated by the constant ¯z and is thus a plane. For minimal surfaces this classically implies that ¯x∈∂^∗E.

Remark 4.8. In dimension2 and 3, the converse is also true (see [22]) meaning that calibrations have Lebesgue points at every regular point of a calibrated set.

Thanks to Proposition 3.4, one can order the minimizers which are calibrated by a given vector field.

Proposition 4.9. Letz∈X calibrates two plane-like minimizersE1 andE2 with connected boundaries. Then, either E1⊆E2, orE2⊆E1. As a consequenceH^d−3(∂E1∩∂E2) = 0.

Proof. Ifz calibratesE1 andE2then it also calibratesE1∩E2 andE1∪E2 which are then also Class A minimizers by Theorem 4.2. SinceE1∩E2 ⊆ E1, by Proposition 3.4, either E1∩E2 =E1 in which case E1 ⊆E2, or H^d−3(∂(E1∩E2)∩∂E1) = 0 which implies that E2⊆E1.

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4.1 Calibrations and the Birkhoff property

We now define the class of plane-like minimizers that we are going to consider in the analysis of the differentiability properties of ϕ. If E = {x : vp(x) +p·x > s} for somevp which minimizes (6), we have that E+q = {vp(x) +p·x > s+p·q} for all q ∈ Z^d, therefore E=E+qifp·q= 0,E⊆E+qifp·q <0, andE+q⊇E ifp·q >0. This is called the Birkhoff property. Notice that in this case we also haveE=S

{q·p>0, q∈Zd}(E+q).

Definition 4.10. Following [27, 37, 12] we give the following definitions:

• we say thatE⊆R^d satisfies the Birkhoff property if, for any q∈Z^d, eitherE⊆E+q orE+q⊆E;

• we say thatE satisfies the strong Birkhoff property in the directionp∈Z^d ifE⊆E+q whenp·q≤0andE+q⊆E whenp·q≥0;

• we say that a plane-like minimizerEin the directionpis recurrent if eitherpis rational andE has the strong Birkhoff property, or if

E= [

q·p>0,q∈Zd

(E+q) or E= \

q·p<0,q∈Zd

(E+q). (19)

Remark 4.11. Observe that if E satisfies the Birkhoff property, there existsp∈R^d such that if q ∈ Z^d, q·p > 0, then E+q ⊆ E, while E+q ⊇ E if q·p < 0 (the difference with the strong Birkhoff property is in the fact that whenq·p= 0, then one might not have E+q=E). The vectorp(up to multiplication with a positive scalar) is uniquely determined, unlessE+q=E for allq∈Z^d. See [12], or Lemma A.1 in Appendix A for an elementary proof of this claim.

Remark 4.12. A recurrent plane-like minimizer always enjoys the strong Birkhoff property (since the setS

q·p>0(E+q), for instance, obviously does).

We will letCA(p) be the set of all the plane-like minimizers in the directionpwhich satisfy the strong Birkhoff property.

The following result can be deduced from [21].

Lemma 4.13. IfE is a Class A minimizer which satisfies the Birkhoff property, andE6=∅, E6=R^d, then it is plane-like in the direction given by Remark 4.11. Moreover it satisfies (5) with a constantM depending only on the anisotropy F and the dimension d.

Proof. All the arguments can be found in the proofs of Proposition 8.3, Proposition 8.4 and Lemma 8.5 in [21]. First, if for somea∈R^d,a+ [0,1)^d⊆E, thenE contains the half-space {x·p > a·p+P

i|pi|} ⊆S

q·p>0(q+a) + [0,1)^d, and similarly, if (a+ [0,1)^d)∩E=∅, then E is contained in a half-space.

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Assume for instance that E does not contain a half-space, hence that a+ [0,1)^d∩E^c 6=∅ for all a∈R^d. Then, by the density estimate, |E^c∩(a+ [−1/2,3/2)^d)| > δ >0 for anya and some constantδwhich depends only onc0and the dimensiond. Now, we also have that b+ [0,1)^d∩E 6= ∅ for some b ∈ R^d, otherwise E would be empty. Then, for anyq ∈ Z^d withq·p≥0, (q+b+ [0,1)^d)∩E ⊆(q+b+ [0,1)^d)∩(q+E)6=∅. Again it follows that

|(q+b+ [−1/2,3/2)^d)∩E| ≥δ >0. We deduce that the energyR

q+b+[−1/2,3/2)^dF(x, DχE) is bounded from below, by some constantδ^′ >0. Hence, if BR(xR) is a large ball contained in {x : (x−b)·p≥0}, the energy in the ball is bounded below byN δ^′, whereN := #{q∈Z^d : q+b+ [−1/2,3/2)^d⊆BR(xR)} ∼=R^d. However, by Class A minimality it is also less than c⁻¹₀ dωdR^d−1, a contradiction. It follows thatEsatisfies (5), with a constantM independent onE.

Proposition 4.14. Let E be a Class A Minimizer with the Birkhoff property, then E has a periodic calibration.

Proof. LetR >0 andk≥1, and let vk(x) := X

q∈Zd,|q|≤k

χE+q ∈ BV(BR)

where in the sum, we drop the terms which are 1 a.e. on BR. Thanks to the Birkhoff property, the setsE+q,|q| ≤k are exactly the level sets ofvk. Consider nowv∈BV(BR) such thatv−vkhas support inBR. Fors∈R, the level set{v > s}is a compactly supported perturbation of the level set{vk > s}. Since this latter set is a Class A Minimizer, one has

Z

BR

F(x, Dχ{vk>s}) ≤ Z

BR

F(x, Dχ{v>s}). (20) Hence,

Z

BR

F(x, Dv) = Z ∞

−∞

Z

BR

F(x, Dχ_{v>s})ds

≥ Z ∞

−∞

Z

BR

F(x, Dχ_{v_k_>s})ds = Z

BR

F(x, Dvk), (21) so thatvk is minimizing inBR, with its own boundary datum. This yields the existence of a calibrationz_k^R∈L^∞(BR;R^d), such thatF^◦(x, z_k^R(x))≤1 a.e., divz_k^R= 0, and [z_k^R, Dvk] = F(x, Dvk) (in the sense of measures). By construction, the latter property is equivalent to [z_k^R, DχE+q] =F(x, DχE+q) for anyq∈Z^d with|q| ≤k, that is,z^R_k is also a calibration for each minimizing setE+q, inside BR.

Now, we letk → ∞: up to a subsequence, z_k^R will converges, weakly-∗ in L^∞(BR;R^d), to somezRwhich will be a calibration for all the setsE+q,q∈Z^d, in the ballBR(cf Prop. 4.3).

Then we can sendR → ∞, in that casezR (extended by zero out ofBR) converges again, weakly-∗ in L^∞(R^d;R^d), to some z which is a calibration for all the setsE+q,q ∈Z^d, in

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any ball. Let us now show thatz may be chosen to be periodic: indeed, clearly,z(x−q) is also a calibration forE and all its translates, for anyq∈Z^d. One may consider for anyk

z_k^′(x) := 1

#{q∈Z^d : |q| ≤k} X

q∈Zd,|q|≤k

z(x−q). (22)

which again, will be a calibration forEand all its translates. Passing to the limit, it converges (up to subsequences) to a new calibrationz^′, which is now periodic.

Proposition 4.15. LetE be a plane-like minimizer with a periodic calibration, then E has the Birkhoff property and∂E is connected.

Proof. Without loss of generality we can assume that E satisfies (5) with a = 0. Since E has a periodic calibration, by Theorem 4.2 it is a Class A Minimizer. Moreover, since every connected component E is also calibrated, any of them is a Class A Minimizer. Let E0 be the connected component of E which contains the half space {x·p > M} and let E1 be another connected component of E. By Proposition 3.1, for every R > 0 we have E(E1, BR)≥c0βR^d−1. However, sinceE1⊆ {|x·p| ≤M}, the minimality ofE1 also yields E(E1, BR)≤CM R^d−2. This is a contradiction ifR is large enough, and it follows thatE is connected.

An analogous argument gives that E^c is also connected, thus implying the thesis. The Birkhoff property is deduced from Proposition 4.9, applied toEandq+E,q∈Z^d.

From Lemma 4.13 and Propositions 4.14 and 4.15 we obtain the following

Corollary 4.16. Let E be a Class A Minimizer with the Birkhoff property, then ∂E is connected.

Remark 4.17. An interesting question raised by Bangert in [13] for non-parametric integrands is whether every plane-like minimizer necessarily satisfies the Birkhoff property. In [13, Th. 8.4], Bangert proves that, in the non-parametric case, it is true for totally irrational vectors p. Propositions 4.14 and 4.15 show that, in the parametric case, this question is equivalent to understand if every plane-like minimizer has a periodic calibration. See also [28] where a nice relation is given between this question of Bangert and De Giorgi’s conjecture.

We also show the following result:

Proposition 4.18. E is a recurrent plane-like minimizer in the direction pif and only if there exists a minimizervp of (6) such that

E={x : vp(x) +p·x >0} or E={x : vp(x) +p·x≥0}. Proof. The “if part” is straightforward, as already observed.

IfE is a recurrent plane-like minimizer (hence with the strong Birkhoff property), by Propo- sition 4.14E has a periodic calibrationz. We build a function vk, k ≥1, as follows: since

(22)

E has the strong Birkhoff property, we can define in BVloc(R^d) a function vk such that {vk ≥p·q}=E+qfor allqwith|q| ≤k. Indeed,Ebeing plane-like, ifp·q >0 one cannot haveE+q=E, otherwise repeating the translation one would reach a contradiction. Hence the function

vk(x) := sup{p·q : |q| ≤k, x∈E+q} (23) has actually the right level sets. Now, sinceE is plane-like, its oscillation is also uniformly bounded, and in fact it is locally uniformly bounded inL^∞. By construction,zis a calibration forvk, which means in particular that for anyR >0,

Z

BR

F(x, Dvk) = Z

BR

[z, Dvk] = − Z

∂BR

vk[z, ν^B^R] ≤ 2^dCRc⁻¹₀

where CR is a uniform bound for kvkkL^∞(BR). Hence the vk are uniformly bounded in BV(BR): up to a subsequence, we may assume thatvk→v inL¹_loc(R^d) withv∈BVloc(R^d).

If|q| ≤k, thenE+q⊆ {vk≥p·q}. Passing to the limit it follows thatE+q⊆ {v≥p·q}. Conversely, if (E+q)^c⊆ {vk< p·q}. Hence (E+q)^c⊆ {vk≤p·q}. Then, for anyq∈Z^d, {v > p·q} ⊆E+q⊆ {v≥p·q}. (24) If pis rational, then it is obvious that equality holds in (24), since, in fact, one can check thatvk does not change whenkis large enough. Ifpis not rational, sinceE is recurrent we can assume thatE=S

p·q>0(E+q), and we shall prove thatE ⊆ {v >0}. This will imply thatE={v >0}, andE+q={v > p·q} for everyq∈Z^d.

Ifx∈E then, for some q∈Z^d withp·q >0, we havex∈E+q and thus, fork ≥ |q|, we also havevk(x)≥p·q >0. Since the sequencevk(x) is increasing, we getv(x)>0 and thus E⊆ {v >0}. The caseE=T

p·q<0(E+q) is similar and givesE={v≥0}.

Let us show thatv−p·xis periodic. It is enough to show that for almost everytand for all q∈Z^d, we have{v≥t}+q⊆ {x : v≥t+p·q}. Then, lettingvp(x) =v(x)−p·x, we will deduce that

vp(x)≥t−p·x =⇒ vp(x+q)≥t+p·q−p·(x+q) =t−p·x

for almost every t, x and for all q ∈ Z^d, yielding that vp is periodic (indeed, being Z^d countable, the converse also holds for a.e.tandx).

For a.e. t we have Et = {v ≥ t} = {v > t}. In that case, Et is the Kuratowski limit of E+qn for some sequence (qn) in Z^d, with p·qn → t as n → +∞. In particular, Et is calibrated by z. Now, for k large enough and fixed n, E+qn+q = {vk ≥ p·(qn +q)}, hence in the limitE+qn+q⊆ {v ≥t+p·q}. Passing then to the limit innwe find that {v≥t}+q⊆ {v≥t+p·q}, which is our claim.

As already observed, almost all level sets ofvare calibrated byz. It follows easily thatvpis a minimizer of (6).

(23)

An example of a non recurrent plane-like minimizer satisfying the strong Birkhoff property would be a setE with gaps on boths side of its boundary, however we do not know whether this situation can occur.

Thanks to Proposition 4.18 we can now prove that every calibration calibrates every plane- like minimizer with the strong Birkhoff property, which implies that CA(p) is stable under union or intersection.

Theorem 4.19. Let z be a calibration in the direction p, then z calibrates every plane-like minimizer with the strong Birkhoff property.

Proof. By Proposition 4.18 we know that every recurrent plane-like minimizer is of the form {vp+p·x > s} or {vp+p·x ≥ s}, for some periodic function vp minimizing (6), which implies that it is calibrated by every calibration.

We can thus assume that p is irrational and that we are given a non recurrent plane-like minimizerE, but which satisfies the strong Birkhoff property. Then, the setEe:=S

q·p>0(E+

q) is a plane-like recurrent Class A Minimizer (Proposition 3.2), hence calibrated by z.

Moreover, it satisfies

Ee⊆E and E⊆Ee+q ∀q∈Z^d, q·p <0. (25) Using the notation of Section 2.1, we define the quotient torus with respect to the rational directions orthogonal top,T_r:=R^d/Γ(p).

If we are given a calibrationz in the direction p, since it is periodic and since E andEe are invariant by translations in Γ(p), we can identify them with their equivalence class inT_r. Thanks to (25), the measure |E\Ee| is then finite. Reminding that V^r(p) = SpanRΓ(p), we let for every x ∈ R^d, f(x) := minx^r∈V^r(p)|x−x^r|. Then, the projection of f on T_r (still denoted by f) is well defined, and satisfies |∇f| = 1 a.e. in T^r. Given s ∈ R we let Cs:={x∈T_r : f(x)≤s}. Then, it follows (from the Coarea Formula) that

|E\Ee| = Z ∞

0 H^d−1((E\E)e ∩∂Cs)ds . Hence, there exists a sequencesi→+∞such that

H^d−1((E\E)e ∩∂Csi)≤ 1 si

. (26)

Now, Proposition 4.14 shows that every plane-like minimizer which satisfies the Birkhoff property is calibrated by a periodic calibration, and we can easily deduce that its projection in T_r is also minimizing (with respect to compact perturbation inside the cylinder). In particular, comparing the energy of E with the energy of (E\Csi)∪(Ee∩Csi), it follows from (26)

Z

∂^∗E∩C_si

F(x, ν^E)≤ Z

∂^∗E∩Ce _si

F(x, ν^E^e) +c⁻¹₀ si

. (27)